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\begin{abstract}

The objective of this study was to model the volume expansion factor ($VEF$), defined as being the ratio between the total aboveground woody volume and the stem merchantable volume of a tree, as a function of tree height and diameter at breast height. 

A large database (dataset \#1) of detailed stem and branches volume measurements, constituted by 8192 felled trees from 19 temperate tree species, was used for calibrating the models. In addition, an independent dataset (dataset \#2), constituted by 176 felled trees from 13 species, was collected for validating the models.

From dataset \#1, the RMSE for the prediction of the total volume varied from 0.005 to 0.476 $m^{3}$, depending on the diameter class, and the corresponding relative RMSE varied from 8.0\% to 13.7\%, depending on the diameter class. 

A ten-fold cross-validation on dataset \#1 gave an average RMSE of 0.136 for the prediction of the $VEF$ and of 0.150 $m^{3}$ for the prediction of the total volume, which was in the same order as average RMSE obtained from the whole dataset \#1.  

Validation on dataset \#2 gave satisfactory results, even for the application of $VEF$ to other tree species: RMSE obtained for the different tree species were in the same order as RMSE obtained from calibration dataset \#1, except for some particular cases. The largest errors were obtained when the model was clearly used in extrapolation, e.g., for five large-size \textit{Fraxinus} (whereas dataset \#1 included smaller trees) and several \textit{Quercus} and \textit{Fagus} from coppice-with-standards stands (whereas dataset \#1 included mainly pure even-aged high-forest trees).

The observed differences between species seemed consistent with the general knowledge about species-specific traits. For a given diameter at breast height, angiosperms were found to have a much larger volume of branches in comparison with the corresponding stem volume than gymnosperms. For a tree of 30 cm in diameter, the lowest values of $VEF$ were obtained for \textit{Picea} and \textit{Abies} ($VEF < 1.1$) and the highest ones for \textit{Fraxinus} and \textit{Carpinus} ($VEF > 1.3$).

The methodology that was developed, based on nonlinear mixed-effects modeling, is easily applicable to other definitions of $VEF$ or biomass expansion factors. 
\end{abstract}


\begin{keyword}
Volume \sep Biomass \sep VEF \sep BEF \sep Nonlinear mixed-effects model

\end{keyword}
